Spectral relations for a matrix model in fermionic Fock space

We consider a matrix model ${\mathcal A}$, related to a system describing two identical fermions and one particle of another nature on a lattice, interacting via annihilation and creation operators. The problem of the study of the spectrum of a block operator matrix ${\mathcal A}$ is reduced to the investigation of the spectrum of block operator matrices of order three with a discrete variable, and relations for the spectrum, essential spectrum and point spectrum are established. Two-particle and three-particle branches of the essential spectrum of the block operator matrix ${\mathcal A}$ are singled out.